For each n≥0n \geq 0, the nthn^{th}symmetric powerX↦Sn(X)X \mapsto S^n(X) is a Schur functor.

For each n≥0n \geq 0, the nthn^{th}alternating powerX↦Λn(X)X \mapsto \Lambda^n(X) is a Schur functor.

More generally, complex irreducible representations of SnS_n correspond to nn-box Young diagrams, so Schur functors are usually described with the help of these. An nn-box Young diagram is simply a pictorial way of writing nn as a sum of natural numbers listed in decreasing order. For example, this 17-box Young diagram:

describes the partition of 1717 as 5+4+4+2+1+15 + 4 + 4 + 2 + 1 + 1. However, it also can be used to construct an irreducible complex representation of the permutation group S17S_{17}, and thus a Schur functor.

In general, here is the recipe for constructing the Schur functor SλS_\lambda associated to an nn-box Young diagram λ\lambda. For now we only say what this functor does to objects (that is, finite-dimensional vector spaces):

Given an nn-box Young diagram λ\lambda and a vector space XX, we first form the tensor power X⊗nX^{\otimes n}. We think of each factor in this tensor power as corresponding to a specific box of the Young diagram.

Then we pick out the subspace of X⊗nX^{\otimes n} consisting of tensors that are unchanged by any permutation that interchanges two boxes in the same row.

Then we project this subspace onto the space of tensors that change sign under any permutation that interchanges two boxes in the same column. The result is called Sλ(X)S_\lambda(X), where SλS_\lambda is the Schur functor corresponding to λ\lambda.

It is easy to see from this description that:

The tall skinny Young diagrams with one column and nn rows give the Schur functors Λn\Lambda^n.

The short fat Young diagrams with one row and nn columns give the Schur functors SnS^n.

We can also think of this relation between Young diagrams and Schur functors in a slightly more abstract way using the group algebra of the symmetric group, ℂ[Sn]\mathbb{C}[S_n]. Suppose λ\lambda is an nn-box Young diagram. Then we can think of the operation ‘symmetrize with respect to permutations of the boxes in each row’ as an element pλS∈ℂ[Sn]p^S_\lambda \in \mathbb{C}[S_n]. Similarly, we can think of the operation ‘antisymmetrize with respect to permutations of the boxes in each column’ as an element pλA∈ℂ[Sn]p^A_\lambda \in \mathbb{C}[S_n]. By construction, each of these elements is idempotent:

Now, it is easy to see that the product of commuting idempotents is idempotent. The elements pλSp^S_\lambda and pλAp^A_\lambda do not commute, but amazingly, their product

pλ=pλApλS p_\lambda = p^A_\lambda p^S_\lambda

is still idempotent!

This element pλ∈ℂ[Sn]p_\lambda \in \mathbb{C}[S_n] is called the Young symmetrizer corresponding to the nn-box Young diagram λ\lambda. There is a functor, called a Schur functor:

Sλ:FinVect→FinVect S_\lambda : FinVect \to FinVect

defined on any finite-dimensional vector space XX as follows:

Sλ(X)=pλX⊗n S_\lambda(X) = p_\lambda X^{\otimes n}

Here we are using the fact that SnS_n, and thus its group algebra, acts on X⊗nX^{\otimes n}. Thus, pλp_\lambda acts as an idempotent operator on X⊗nX^{\otimes n}, and Sλ(X)S_\lambda(X) as defined above is the range of this operator.

An even deeper approach to Schur functors is based on the relation between Young diagrams and representations of the symmetric groups. The subspace

has an obvious right action of the algebra ℂ[Sn]\mathbb{C}[S_n], and thus becomes a representation of the group SnS_n. In fact, it is an irreducible representation of SnS_n, and every irreducible representation of SnS_n is isomorphic to one of this form. Even better, this recipe sets up a one-to-one correspondence between nn-box Young diagrams and isomorphism classes of irreducible representations of SnS_n.

(Strictly speaking, to think of pλp_\lambda as an element of ℂ[Sn]\mathbb{C}[S_n], we must choose a way to label the boxes of the Young diagram with numbers 1,…,n1, \dots, n. Such an identification is called a Young tableau. For our purposes it will cause no harm to randomly choose a Young tableau for each Young diagram, since different choices give isomorphic representations VλV_\lambda. In other contexts, the difference between various choices of Young tableaux can be extremely important.)

This description of irreducible representations of SnS_n paves the way towards an important generalization of Schur functors. First, note that

Even more generally, any finite direct sum of the Schur functors just described may also be called a Schur functor. In other words, if R={Rn}n≥0R = \{R_n\}_{n \ge 0} where RnR_n is a finite-dimensional representation of SnS_n, and only finitely many of these representations are nonzero, then we define the Schur functor

each RnR_n is a finite-dimensional representation of SnS_n, and only finitely many of these representations are nonzero.

natural transformations between such functors as morphisms.

In the rest of this article, we would like to give a conceptual explanation of this category.

As a warm-up, let us note that Schur\Schur has has a nice description in terms of groupoid of finite sets and bijections. This groupoid is the core of the category FinSet, so it is denoted core(FinSet)core(FinSet). What is the relation between Schur functors and this groupoid. Every Schur functor is a finite direct sum of Schur functors coming from irreducible representations of symmetric groups SnS_n for various nn. But what sort of entity is a direct sum of representations of symmetric groups SnS_n for various nn? It is nothing but a representation of the permutation groupoid:

ℙ=⨆n≥0Sn, \mathbb{P} = \bigsqcup_{n \ge 0} S_n \, ,

where objects are natural numbers, all morphisms are automorphisms, and the automorphisms of nn form the group SnS_n. In other words, it is a functor

R:ℙ→VectR : \mathbb{P} \to Vect

But conceptually, the importance of ℙ\mathbb{P} is that it is a skeleton of the groupoid of finite sets and bijections. So,

ℙ≃core(FinSet). \mathbb{P} \simeq core(FinSet) \, .

As a result, any Schur functor gives a functor

R:core(FinSet)→Vect R : core(FinSet) \to Vect

Following Joyal’s work on combinatorics, such functors are known as VectVect-valued species, or VectVect-valued structure types. The idea here is that just as an ordinary Set-valued species

T:core(FinSet)→Set T : core(FinSet) \to Set

assigns to any finite set a set of structures of some type, a VectVect-valued species assigns to any finite set a vector space of structures of some type, Thanks to the ‘free vector space on a set’ functor

F:Set→Vect,F: Set \to Vect \, ,

we can linearize any ordinary species TT and obtain a linear species F∘TF \circ T. This process is extremely important in the work of Marcelo Aguiar and Swapneel Mahajan:

However, not all VectVect-valued species corresond to Schur functors, because we have defined Schur functors to arise from finite direct sums of irreducible representations of permutation groups. So, SchurSchur is equivalent to the category where:

objects are polynomial species: that is, functors R:core(FinSet)→FinVectR: core(Fin\Set) \to FinVect such that R(n)={0}R(n) = \{0\} for all sufficiently large finite sets nn;

morphisms are natural transformations.

We call this the category of polynomial species. The reason for the term ‘polynomial’ is that any functor of the form

and this power series is a polynomial precisely when RR is a polynomial species.

The category of representations of any groupoid has many nice features. For example, it is a symmetric monoidalabelian category, by which we mean a symmetric monoidal category that is also abelian, where tensoring with any object is right exact. So, the category of VectVect-valued species is symmetric monoidal abelian — and it is easy to check that the subcategory of polynomial species inherits this structure. Since SchurSchur is equivalent to the category of polynomial species, it too is symmetric monoidal abelian.

In particular, SchurSchur has two monoidal structures, ⊕\oplus and ⊗\otimes, defined by

(F⊕G)(V)=F(V)⊕G(V) (F \oplus G)(V) = F(V) \oplus G(V)

and

(F⊗G)(V)=F(V)⊗G(V) (F \otimes G)(V) = F(V) \otimes G(V)

Since ⊗\otimes distributes over ⊕\oplus, these make SchurSchur into a rig category.

In the literature on species, the operation ⊕\oplus is often called addition, since adding species this way corresponds to adding their generating functions. Aguiar and Mahajan call ⊗\otimes the Hadamard product.

But the category of representations of a groupoid has even more nice features when the groupoid itself has a monoidal structure: then the representation category acquires a monoidal structure thanks to Day convolution. The groupoid ℙ\mathbb{P} has, in fact, two important monoidal structures, coming from the product ×\times and disjoint union ++ of finite sets. Since ×\times distributes over ++, these make ℙ\mathbb{P} into a rig category. Thanks to Day convolution, these give the category of representations of ℙ\mathbb{P} two more monoidal structures, making it into a rig category in another way. The same is true for the subcategory SchurSchur.

In the literature on species, the monoidal structure coming from ++ is often called multiplication, since multiplying species in this way corresponds to multiplying their generating functions. Aguiar and Mahajan call this monoidal structure the Cauchy product. The monoidal structure coming from ×\times has no commonly used name, but it deserves to be called the Dirichlet product.

Aguiar and Mahajan point out that some of the relationships between the “pointwise” products (addition and Hadamard) and the “Day” products (Cauchy and Dirichlet) can be described in terms of duoidal categories. Specifically, the Hadamard and Cauchy products form duoidal structures in both orders.

On top of all this, the composite of Schur functors is again a Schur functor. This gives SchurSchur a fifth monoidal structure: the plethystic tensor product. Unlike the four previous monoidal structures, this one is not symmetric.

Schur functors on more general categories

We have described Schur functors as special functors

F:FinVect→FinVect F: FinVect \to FinVect

But in fact, functors such as the nthn^{th} alternating power, nthn^{th} symmetric power, etc. make sense in much wider contexts. For starters, we can replace the complex numbers by any field kk of characteristic zero, and everything in our discussion still works. More importantly, Schur functors can be applied to any symmetric monoidalCauchy completelinear category. Here by linear category we mean a category enriched over VectVect, the category of vector spaces over kk. Such a category is Cauchy complete when:

the category of coherent sheaves of vector spaces over any algebraic variety (or scheme or algebraic stack) over kk

These examples can be hybridized, and thus they multiply indefinitely: for example, we could take coherent sheaves of chain complexes, or vector bundles equipped with a group action, and so on.

In the following subsections, we explain how to define Schur functors on any category of this sort. A somewhat novel feature of our treatment is that we do not require the theory of Young diagrams to define and study Schur functors.

Our strategy is as follows. We fix a symmetric monoidal Cauchy complete linear category, CC. The group algebra k[Sn]k[S_n] begins life as a monoid in the symmetric monoidal category FinVectFinVect. However, we shall explain how interpret it as living in CC by a “change of base” functor going from FinVectFinVect to CC. This will let us use the Young symmetrizers pλp_\lambda to construct idempotents on V⊗kV^{\otimes k} for any object V∈CV \in C. Splitting these idempotents, we obtain the Schur functors Sλ:C→CS_\lambda : C \to C.

Change of base

To achieve the desired change of base, let MatMat be the linear category whose objects are integers m≥0m \geq 0 and whose morphisms m→nm \to n are m×nm \times n matrices with entries in kk. Because CC is Cauchy complete and in particular has finite biproducts (direct sums), there is an evident linear functor

Mat→CMat \to C

which takes mm to ImI^m, the direct sum of mm copies of the tensor unit II. It is the unique linear functor taking 11 to II, up to unique linear isomorphism. In the case C=FinVectC = FinVect, the linear functor

Mat→FinVect,Mat \to FinVect,

taking 11 to kk, is a linear equivalence (exhibiting MatMat as a skeleton of CC). Because of this equivalence, we could equally well say that there is a linear functor

i:FinVect→Ci: FinVect \to C

which, up to unique linear isomorphism, is the unique linear functor taking kk to II. Notice that a symmetric monoidal functor of this form must take the tensor unit kk to II (up to coherent isomorphism, as always), and in fact ii is symmetric monoidal, because there is a canonical isomorphism

Im⊗In≅Imn,I^m \otimes I^n \cong I^{m n},

using the fact that ⊗\otimes preserves direct sums in each argument, and the fact that there is a canonical isomorphism I⊗I≅II \otimes I \cong I.

Proposition

The action of Young symmetrizers

Next we explain how given an object X∈CX \in C, any Young symmetrizer in k[Sn]k[S_n] acts as an idempotent on X⊗nX^{\otimes n}.

For this we only need to know a little bit about the group algebra k[Sn]k[S_n], which we recall here. By Maschke's theorem, for any finite group GG, the group algebrak[G]k[G] decomposes as a direct sum of matrix algebras

⨁λhom(Vλ,Vλ)\bigoplus_{\lambda} hom(V_\lambda, V_\lambda)

where λ\lambda ranges over isomorphism classes of irreducible representations of GG. The identity elements of these matrix algebras hom(Vλ,Vλ)hom(V_\lambda, V_\lambda) thus correspond to certain special elements pλ∈k[G]p_\lambda \in k[G]. Clearly these elements are idempotent:

pλ2=pλ. p_\lambda^2 = p_\lambda \, .

We are particularly interested in the case G=SnG = S_n. In this case, we call the idempotents pλp_\lambda are ‘Young symmetrizers’. However, we will not need the formula for these idempotents.

The key step is to apply base change to k[Sn]k[S_n]. Here we exploit the fact that

k[Sn]=⨁σ∈Snk k[S_n] = \bigoplus_{\sigma \in S_n} k

is a monoid in the monoidal category FinVectFinVect. Since i:FinVect→Ci : FinVect \to C is a monoidal functor, it follows that ii carries k[Sn]k[S_n] to a monoid in CC, which we again call k[Sn]k[S_n] by abuse of notation. As an object of CC, we have

(1)k[Sn]≅⨁σ∈SnI k[S_n] \cong \bigoplus_{\sigma \in S_n} I

There is a general concept of what it means for a monoid in a monoidal category to act on an object in that category. In particular, if XX is an object of CC, the monoid k[Sn]k[S_n] acts on the tensor power X⊗nX^{\otimes n}. To see this, note that for each σ∈Sn\sigma \in S_n, there is a corresponding symmetry isomorphism

Finally, we would like to describe how each Young symmetrizer pλ∈k[Sn]p_\lambda \in k[S_n] acts on X⊗nX^{\otimes n}. Quite generally, any element x∈k[Sn]x \in k[S_n] gives a linear map from kk to k[Sn]k[S_n], namely the unique map sending 11 to xx. Applying the functor ii to this, we obtain a morphism which by abuse of language we call

Constructing Schur functors

By construction, the morphisms

p˜λ:X⊗n→X⊗n \widetilde{p}_\lambda : X^{\otimes n} \to X^{\otimes n}

are the components of a natural transformation from the functor X↦X⊗nX \mapsto X^{\otimes n} to itself. Since idempotents split in CC, we can form the cokernel of 1−p˜λ1 - \widetilde{p}_\lambda, or in other words, the coequalizer of the pair

Definition

For any Young diagram λ\lambda, the Schur functorSλ:C→CS_\lambda: C \to C is defined as follows. Given an object XX of CC, let Sλ(X)S_\lambda(X) be the cokernel of p˜λ:X⊗n→X⊗n \widetilde{p}_\lambda : X^{\otimes n} \to X^{\otimes n}. Given a morphism f:X→Yf: X \to Y in CC, let Sλ(f)S_\lambda(f) be the unique map Sλ(X)→Sλ(Y)S_\lambda(X) \to S_\lambda(Y) such that

for any finite-dimensional representation RR of SnS_n, as follows. We can write RR as a finite direct sum of irreducible representations:

R=⨁iVλi R = \bigoplus_i V_{\lambda_i}

and then define

SR(−)=⨁iSλi(−). S_R(-) = \bigoplus_i S_{\lambda_i}(-) \, .

Schur functors are “natural”

Suppose now that we have a symmetric monoidal linear functor G:C→DG: C \to D. We can think of GG as a “change of base category”, and Schur functors are “natural” with respect to change of base.

That is to say: if GG is a symmetric monoidal linear functor C→DC \to D, then by definition GG preserves tensor products (at least up to coherent natural isomorphism), and GG will automatically preserve both direct sums (by linearity) as well as splittings of idempotents (as all functors do). Therefore, for a Schur functor Sλ(X)=Vλ⊗SnX⊗nS_\lambda(X) = V_\lambda \otimes_{S_n} X^{\otimes n}, we have natural isomorphisms

where the first isomorphism uses the symmetric monoidal structure of GG; the second uses the fact that Vλ,D≅G(Vλ,C)V_{\lambda, D} \cong G(V_{\lambda, C}) because there is, up to isomorphism, only one symmetric monoidal linear functor FinVect→DFinVect \to D; the third uses the symmetric monoidal structure again and preservation of idempotent splittings.

If RR is any representation, then by writing RR as a direct sum of irreducible representations VλV_\lambda and using the fact that GG preserves direct sums, we have more generally

Conceptual description of Schur functors

As we have seen, Schur functors SRS_R are definable under fairly mild hypotheses: working over a field of characteristic zero, they can be defined on any symmetric monoidal linear category CC which is Cauchy complete. So, for such CC we can define a Schur functor

SR:C→CS_R: C \to C

and moreover, if G:C→DG: C \to D is a symmetric monoidal linear functor, the Schur functors on CC and DD are “naturally” compatible, in the sense that the diagram

In this abstract framework, it may be wondered what significant role is played by the representations RR of the symmetric group. The natural isomorphisms ϕG\phi_G which relate the Schur functors across change of base G:C→DG: C \to D are pleasant to observe, but surely this is just some piddling general nonsense in the larger story of Schur functors SRS_R, which are after all deeply studied and incredibly rich classical constructions?

Let us put the question another way. We have seen the Schur functors SRS_R are constructed in a uniform (or “polymorphic”) way across all symmetric monoidal Cauchy complete linear categories CC, and this construction is natural with respect to symmetric monoidal change of base functors G:C→DG: C \to D. Or rather: not natural in a strict sense, but pseudonatural in the sense that naturality squares commute up to isomorphism ϕG\phi_G. Now pseudonaturality is a very general phenomenon in 2-category theory. So the question is: among all such pseudonatural transformations SS, what is special about the Schur functors SRS_R? What extra properties pick out exactly the Schur functors SRS_R from the class of all pseudonatural transformations SS?

The perhaps surprising answer is: no extra properties! That is, the Schur functors SRS_R are precisely those functors that are defined on all symmetric monoidal Cauchy complete linear CC and that are pseudonatural with respect to change of base G:C→DG: C \to D!

Let us now make this precise. Schur functors are defined on certain symmetric monoidal linear categories, but they respect neither the symmetric monoidal structure nor the linear structure. So, we have to forget some of the structure of the objects on which Schur functors are defined. This focuses our attention on the ‘forgetful’ 2-functor

As we shall see, Schur functors correspond to pseudonatural transformations from UU to itself, and morphisms between Schur functors correspond to modifications between these pseudonatural transformations. For the reader unaccustomed to these 2-categorical concepts, we recall:

Definition

Given two 2-functors U,V:S→→CU, V: S \stackrel{\to}{\to} C between 2-categories, a pseudonatural transformationϕ:U→V\phi: U \to V is a rule that assigns to each 0-cell ss of SS a 1-cell ϕ(s):U(s)→V(s)\phi(s): U(s) \to V(s) of CC, and to each 1-cell f:r→sf: r \to s of SS an invertible 2-cell ϕ(f)\phi(f) of CC:

Definition

With notation as above, let ϕ,ψ:U→V\phi, \psi: U \to V be two pseudonatural transformations. A modificationx:ϕ→ψx: \phi \to \psi is a rule which associates to each 0-cell ss of SS a 2-cell x(s):ϕ(s)→ψ(t)x(s): \phi(s) \to \psi(t) of CC, such that the following compatibility condition holds:

We now propose our conceptual definition of Schur functor:

Definition

An (abstract) Schur functor is a pseudonatural transformation S:U→US: U \to U, where

U:SymMonLinCauch→CatU: SymMonLinCauch \to Cat

is the forgetful 2-functor. A morphism of Schur functors is a modification between such pseudonatural transformations.

What this proposed definition makes manifestly obvious is that Schur functors are closed under composition. This provides a satisfying conceptual explanation of plethysm, as we will explore in the next two sections. However, we should first check that this proposed definition gives a category of Schur functors equivalent to the category SchurSchur defined earlier!

Before launching into the proof, it is worth pondering an easier problem where we replace categories by sets, and symmetric monoidal linear Cauchy-complete categories by commutative rings.So, instead of CatCat let us consider SetSet, and instead of SymMonLinCauchSymMonLinCauch let us consider CommRingCommRing. There is a forgetful functor

U:CommRing→Set. U : CommRing \to Set \, .

What are the natural transformations from this functor to itself? Any polynomial P∈ℤ[x]P \in \mathbb{Z}[x] defines such a natural transformation, since for any commutative ring RR there is a function PR:U(R)→U(R)P_R: U(R) \to U(R) given by

PR:x↦P(x) P_R : x \mapsto P(x)

and this is clearly natural in RR. But in fact, the set of natural transformations from this functor turns out to be preciselyℤ[x]\mathbb{Z}[x]. And the reason is that ℤ[x]\mathbb{Z}[x] is the free commutative ring on one generator!

To see this, note that the forgetful functor

U:CommRing→Set U : CommRing \to Set

has a left adjoint, the ‘free commutative ring’ functor

F:Set→CommRing. F : Set \to CommRing \, .

The free commutative ring on a 1-element set is

F(1)≅ℤ[x] F(1) \cong \mathbb{Z}[x]

and homomorphisms from F(1)F(1) to any commutative ring RR are in one-to-one correspondence with elements of the underlying set of RR, since

So, we say F(1)F(1)represents the functor UU. This makes it easy to show that the set of natural transformations from UU to itself is isomorphic to the underlying set of ℤ[x]\mathbb{Z}[x], namely U(F(1))U(F(1)):

Representability

To build a bridge from abstract Schur functors as pseudonatural transformations to the more classical descriptions, we start with the following key result. In what follows we use kℙk \mathbb{P} to denote the ‘linearization’ of the permutation groupoid: that is, the linear category formed by replacing the homsets in ℙ\mathbb{P} by the free vector spaces on those homsets. We use kℙ¯\widebar{k \mathbb{P}} to denote the Cauchy completion of the linearization of ℙ\mathbb{P}. As we shall see, kℙ¯\widebar{k \mathbb{P}} is equivalent to the category of Schur functors. But first:

Theorem

The underlying 2-functor

U:SymMonLinCauch→CatU: SymMonLinCauch \to Cat

is represented by kℙ¯\widebar{k \mathbb{P}}. In other words:

U(−)≃hom(−,kℙ¯) U(-) \simeq hom(-, \widebar{k \mathbb{P}})

Proof (Sketch)

It is well-known that the permutation category ℙ\mathbb{P}, whose objects are integers m≥0m \geq 0 and whose morphisms are precisely automorphisms m→mm \to m given by permutation groups SmS_m, is the representing object for the underlying 2-functor

Finally, let LinCauchLinCauch denote the 2-category of small Cauchy complete linear categories. The linear Cauchy completion gives a 2-reflector (−)¯:Lin→LinCat\widebar{(-)}: Lin \to LinCat which is left 2-adjoint to the 2-embedding i:LinCauch→Lini: LinCauch \to Lin, and again the 2-adjunction (−)¯⊣i\widebar{(-)} \dashv i lifts to the level of symmetric monoidal structure to give a 2-adjunction

For this, the key fact is that if A⊗BA \otimes B denotes the tensor product of two VV-enriched categories, then there is a canonical enriched functor A¯⊗B¯≃A⊗B¯\overline{A} \otimes \overline{B} \simeq \overline{A \otimes B} making Cauchy completion into a lax 2-monoidal functor on VV-CatCat. Even better, it is lax 2-symmetric monoidal. So, it sends symmetric pseudomonoids to symmetric pseudomonoids. In this case, then, it sends symmetric monoidal linear categories to symmetric monoidal linear Cauchy-complete categories.

Putting this all together, the underlying functor U:SymMonLinCauch→CatU: SymMonLinCauch \to Cat is the evident composite

John Baez: I took remarks from the query box here and used them to improve the proof above. However, it could still use more improvement. Could you polish it up a bit, Todd?

Structure of the representing object

Let us now calculate kℙ¯\widebar{k \mathbb{P}}. In general, the linear Cauchy completion of a linear category CC consists of the full subcategory of linear presheaves Cop→VectC^{op} \to Vect that are obtained as retracts of finite direct sums of representables C(−,c):Cop→VectC(-, c): C^{op} \to Vect. In the case C=kℙC = k\mathbb{P}, these are the functors

F:ℙop→FinVectF: \mathbb{P}^{op} \to FinVect

where F(n)=0F(n) = 0 for large enough nn. For it is clear that this category contains the representables and is closed under finite direct sums and retracts. On the other hand, every polynomial FF is a sum of monomials F(0)⊕F(1)⊕⋯⊕F(n)F(0) \oplus F(1) \oplus \cdots \oplus F(n), and by Maschke’s theorem, each SjS_j-module F(j)F(j) is the retract of a finite sum of copies of the group algebra k[Sj]k[S_j] which corresponds to the representable kℙ(−,j)k\mathbb{P}(-, j).

Definition

A polynomial species is a functor F:ℙop→FinVectF: \mathbb{P}^{op} \to FinVect where F(n)=0F(n) = 0 for all sufficiently large nn. A morphism of polynomial species is a natural transformation between such functors.

As we have mentioned, the category of polynomial species inherits two monoidal structures from ℙ\mathbb{P} via Day convolution. Most important is the one coming from the additive monoidal structure on ℙ\mathbb{P}, which is given on the level of objects by adding natural numbers, and on the morphism level given by group homomorphisms

Sm×Sn→Sm+nS_m \times S_n \to S_{m+n}

which juxtapose permutations. This can be linearized to give algebra maps

k[Sm]⊗k[Sn]→k[Sm+n]k[S_m] \otimes k[S_n] \to k[S_{m+n}]

which give the monoidal category structure of kℙk\mathbb{P}. This monoidal structure uniquely extends via Day convolution to the Cauchy completion kℙ¯\widebar{k\mathbb{P}}, which is intermediate between kℙk\mathbb{P} and the category of VectVect-valued presheaves on kℙk\mathbb{P}. The general formula for the Day convolution product applied to presheaves F,G:ℙop→VectF, G: \mathbb{P}^{op} \to Vect is

Theorem

kℙ¯\widebar{k \mathbb{P}} is equivalent to the symmetric monoidal category of polynomial species.

Now, having defined Schur functors abstractly as pseudonatural transformations U→UU \to U, the representability theorem together with the 2-categorical Yoneda lemma means that the category of Schur functors is equivalent to the category of symmetric monoidal linear functors on kℙ¯\widebar{k \mathbb{P}}. Accordingly, we calculate

Theorem

The category SchurSchur is equivalent to the category of polynomial species ℙop→FinVect\mathbb{P}^{op} \to FinVect.

NB: This theorem refers only to the underlying category U(kℙ¯)U(\overline{k\mathbb{P}}). In other words, this category certainly has linear tensor category structure as well, but this structure is not respected by Schur functor composition which we consider next.

Composition of Schur functors

Now we consider composition of Schur functors U→UU \to U, or equivalently symmetric monoidal linear functors kℙ¯→kℙ¯\widebar{k\mathbb{P}} \to \widebar{k \mathbb{P}}. Composition endows [U,U][U, U] with a monoidal structure, and this monoidal structure transfers across the equivalence of the preceding theorem to a monoidal structure on the underlying category of Schur functors, or equivalently, polynomial species ℙop→FinVect\mathbb{P}^{op} \to FinVect. We proceed to analyze this monoidal structure.

It may be easier to do this in reverse. Any Schur functor may regarded as a functor

1→Fkℙ¯.1 \stackrel{F}{\to} \overline{k\mathbb{P}} \, .

This induces a symmetric monoidal functor, unique up to (unique) symmetric monoidal isomorphism:

Here F⊗mF^{\otimes m} is a Day convolution product of mm copies of FF. Finally, the functor F∼F^\sim is linearized and extended (uniquely) to the linear Cauchy completion, to give a symmetric monoidal linear functor on kℙ¯\widebar{k \mathbb{P}}. The efficient tensor product description is

as this manifestly preserves colimits in the blank argument and therefore all colimits needed for the Cauchy completion. (And since the extension to the Cauchy completion is unique, this formula must be correct! The only question is whether this functor is valued in kℙ¯\overline{k\mathbb{P}}.)

In the language of species, this construction is called the substitution product, and is denoted G∘FG \circ F. This is morally correct because it is indeed an appropriate categorification of polynomial composition. However, to avoid overloading the symbol ∘\circ in ways that might be confusing, we will rename it G⊠FG \boxtimes F. Thus,

G⊠F=G⊗ℙF∼G \boxtimes F = G \otimes_{\mathbb{P}} F^\sim

In notation which looks slightly less abstract, this is the Schur object given by the formula

It should be noted that (G⊠F)(n)(G \boxtimes F)(n) is indeed 00 for n>(degG)(degF)n \gt (deg G)(deg F), so that G⊠FG \boxtimes F is indeed a polynomial species. It is just the polynomial special case of the substitution product which is defined on general linear speciesF,G:ℙop→VectF, G: \mathbb{P}^{op} \to Vect.

Proposition

The product ⊠\boxtimes makes the category of polynomial species into a monoidal category. The unit for this product is polynomial species XX given by the representable ℙ(−,1):ℙop→FinVect\mathbb{P}(-, 1): \mathbb{P}^{op} \to FinVect.

Proof (Sketch)

The following proof is adapted from a similar argument due to Max Kelly [ref]: we exhibit an associativity isomorphism α:(−⊠F)⊠G→−⊠(F⊠G)\alpha: (- \boxtimes F) \boxtimes G \to - \boxtimes (F \boxtimes G) on the basis of universal properties. The point is that by the universal property of kℙ¯\overline{k\mathbb{P}}, the category of functors

F:1→kℙ¯F: 1 \to \overline{k\mathbb{P}}

is equivalent to the category of symmetric monoidal linear functors

H:kℙ¯→kℙ¯H: \overline{k\mathbb{P}} \to \overline{k\mathbb{P}}

The correspondence in one direction takes FF to the symmetric monoidal functor H=−⊠FH = - \boxtimes F, and in the other direction takes HH to F=H(X)F = H(X). By the equivalence, we have a unit isomorphism X⊠F≅FX \boxtimes F \cong F. Also by this equivalence, symmetric monoidal linear transformations between symmetric monoidal linear functors of the form

of strong monoidal equivalence from (Schur,⊠)(Schur, \boxtimes) to the monoidal category SymMonLinCauch(kℙ¯,kℙ¯)SymMonLinCauch(\overline{k\mathbb{P}}, \overline{k\mathbb{P}}) under endofunctor composition. (The hexagonal coherence condition for a monoidal functor follows from the pentagon; one side of the hexagon is an identity since endofunctor composition is a strict monoidal product.)

The tensor product ⊠\boxtimes on SchurSchur goes by another name: it is the plethystic tensor product.

References

A nice introduction to Schur functors can be found here:

William Fulton and Joe Harris, Representation Theory: a First Course, Springer, Berlin, 1991.

For a quick online introduction to Young tableaux? and representations of the symmetric groups, try:

Yufei Zhao, Young tableaux and the representations of the symmetric group, MIT. (web)

For more details on these topics, see:

William Fulton, Young Tableaux, with Applications to Representation Theory and Geometry, Cambridge U. Press, 1997.

Old Stuff

The first thing that should be understood from the beginning is that a general Schur functor FF is nonlinear: the action on hom-sets

hom(V,W)→hom(F(V),F(W))hom(V, W) \to hom(F(V), F(W))

is not assumed to respect the linear structure. In fact, linear Schur functors are rather uninteresting: because every finite-dimensional space is a finite direct sum of copies of the 11-dimensional space ℂ\mathbb{C}, and because linear functors preserve finite direct sums (that is, biproducts, it turns out that every linear Schur functor FF is representable as hom(X,−)hom(X, -) where X=F(ℂ)X = F(\mathbb{C}). So, the category of linear Schur functors is equivalent to FinVectFinVect.

John Baez: This stuff should get worked into the discussion near the end of how Schur functors are like polynomials…

Modules over a bimonoid

Next we exploit the fact that, just like any group algebra, k[Sn]k[S_n] is a bialgebra — or in fancier language, a bimonoid in the symmetric monoidal category FinVectkFinVect_k. Since i:FinVectk→Ci : FinVect_k \to C is a symmetric monoidal functor, this means that ii carries k[Sn]k[S_n] to a bimonoid in CC. As noted above, we call this bimonoid by the same name, k[Sn]k[S_n].

The category of modules over a bimonoid is a monoidal category. More explicitly, in the case of the bimonoid k[Sn]k[S_n] in CC with comultiplication

δ:k[Sn]→k[Sn]⊗k[Sn],\delta: k[S_n] \to k[S_n] \otimes k[S_n] \,,

the tensor product V⊗WV \otimes W of two k[Sn]k[S_n]-modules in CC carries a module structure where the action is defined by

where σ\sigma is a symmetry isomorphism and αV\alpha_V, αW\alpha_W are the actions on VV and WW.

Now we consider a particular case of tensor product representations. If XX is an object of CC, the symmetric group SnS_n has a representation on X⊗nX^{\otimes n}. (Indeed, for each σ∈Sn\sigma \in S_n, there is a corresponding symmetry isomorphism X⊗n→X⊗nX^{\otimes n} \to X^{\otimes n}. From this one may construct an action

This operator makes sense since kk has characteristic zero, and crucially, this operator is idempotent (because e=σee = \sigma e for all σ∈Sn\sigma \in S_n). Because we assume idempotents split in CC, we have a (split) coequalizer